C₇×#13573 #13190789

order	1792
nilpotency class	5
isoclinism	isoclinic class of #13569
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₇×#13573

Subgroups

	subgroup	G/H
center, socle	C₂×C₇	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂²×C₄×C₇
Frattini, cosocle	C₄⋊Q₈	C₂³×C₇
prime radical	C₂×C₇×2₋¹⁺⁴	C₄
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂³×C₇ ◁ C₂²×C₄×C₇ ◁ C₂×C₇×C₄⋊Q₈ ◁ G

470 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3,4,5,6,7)(8,9,11,16,13,18,26,36)(10,19,22,30,20,32,38,14)(12,17,28,15,29,33,25,34)(21,37,27,35,24,31,23,39);(8,10)(9,14)(11,24)(12,27)(13,20)(15,32)(16,35)(17,37)(18,30)(19,34)(21,26)(22,25)(23,29)(28,38)(31,33)(36,39);(10,20)(14,30)(19,32)(21,24)(22,38)(23,27)(31,37)(35,39)

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 6561);